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单词 MinimalPolynomial
释义

minimal polynomial


Let K/F be a field extension and κK be algebraic over F. The minimal polynomial for κ over F is a monic polynomial m(x)F[x] such that m(κ)=0 and, for any other polynomialPlanetmathPlanetmath f(x)F[x] with f(κ)=0, m divides f. Note that, for any element κ that is algebraic over F, a minimal polynomial exists (http://planetmath.org/ExistenceOfTheMinimalPolynomial); moreover, because of the monic condition, it exists uniquely.

Given κK, a polynomial m is the minimal polynomial of κ if and only if m(κ)=0 and m is both monic and irreducible (http://planetmath.org/IrreduciblePolynomial).

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更新时间:2025/5/4 18:44:07